# Integrating for Fourier Series

1. Jun 1, 2010

### tomeatworld

1. The problem statement, all variables and given/known data
For positive integers m and n, calculate the two integrals:

$$\frac{1}{L}\int^{L}_{-L}sin(\frac{n \pi x}{L})sin(\frac{m \pi x}{L})dx$$ and $$\frac{1}{L}\int^{L}_{-L}sin(\frac{n \pi x}{L})cos(\frac{m \pi x}{L})dx$$

2. Relevant equations
$$\int u v' dx = [u v] - \int u' v dx$$

3. The attempt at a solution
For the first one, I can't seem to get anything other than 0 for the integral (but not in the normal way). If I work through, I end with $$I = \frac{n^{2}}{m^{2}}I$$ and that makes no sense at all. Every other part of the integral I find cancels to 0 as they all include $$sin(\frac{n \pi x}{L}) or sin(\frac{m \pi x}{L})$$ which will be 0 as n and m are integers. What am I doing wrong?

Edit: I plugged the two integrals into Mathematica and found the second one to be 0, which I calculated, but the first one is not. What do I need to change in my working?

Last edited: Jun 1, 2010
2. Jun 2, 2010

### n.karthick

You could use the relations

$$sinAsinB=\frac{cos(A-B)-cos(A+B)}{2}$$

$$sinAcosB=\frac{sin(A+B)+sin(A-B)}{2}$$

This will make your integration easy.
It is better to view graphically too. Try to plot multiplication of two sine waves of different frequencies ( a sin and cos wave of different frequencies also) and see graphically. You can guess their average value graphically. In fact that is the result of your integration also.